Differentiable functions from the complex numbers to the complex numbers are quite special and are called holomorphic. Holomorphic functions do satisfy the Cauchy-Riemann equations. If the holomorphic function f(z) is written f(x + iy) = u(x + iy) + iv(x + iy) then ¶u/¶x = ¶v/¶y and ¶u/¶y = – ¶v/¶x. One consequence of this is that ¶2u/¶x2 + ¶2u/¶y2 = ¶2v/¶x¶y – ¶2v/¶y¶x = 0. It follows that both u and v satisfy the Laplace equation, Df = ¶2f/¶x2 + ¶2f/¶y2 so Du = ¶2u/¶x2 + ¶2u/¶y2 and Dv = ¶2v/¶x2 + ¶2v/¶y2.

Complex differentiation is a much stronger condition then real differentiation.

Let f be the derivative of the holomorphic function F. The path integral is written as òP f(z) dz. If P is a path that starts at point u and ends at point w then the path integral is well defined, that is unique whatever path is taken in the complex plane and in fact òP f(z) dz = F(w) – F(u). If u and w are the same point then the path integral is equal to zero.

It turns out that the path integral is true if we restrict the domain to be a simply connected subset of C. Simply connected means an open set without holes. If there are holes then the path integrals may differ depending upon how the paths circumnavigate the holes. Path integrals have a close connection with the topology of subsets of the plane.

A holomorphic function can be differentiated an infinite number of times. So for complex functions differentiability implies infinite differentiability.

Any holomorphic function can be expanded in a power series: f(z) = S¥n=0 an(z-w)n , where f is defined and differentiable on an open disk centered at w. This series is called the Taylor series expansion of f.

The enitire behavior of a holomorphic function is determined by their values in a tiny open region. The process of analytical continuation determines the values of the function outside the tiny open region. This process is how the Riemman Zeta function is defined everywhere.

The therorem of Liouville states that if f is a holomorphic function defined on the entire complex plane and f is bounded then f must be constant. A counterexample to this in the real line is the function sin(x), which is not constant but is bounded and infinitely differentiable.

Modern geometry has established two basic ideas: the relationship between geometry and symmetry and the notion of a manifold.

Geometry is concerned with the usual geometrical objects, point, line, plane, space, surve, sphere, cube, distance and angle, as well as refelection, rotation, translation, stretch, shear, projection, angle-preserving map, continuous deformation and transformation. For any group of transformations there is a corresponding notion of geometry, in which one finds phenomena that are unaffected by transformations in that group. Two objects are equivalent if they can be turned into each other from transformations within a group. Tus different groups of transformations lead to different geometries.

Euclidean geometry requires the specification of the dimension, n, and the transformations in the group. The group includes rigid transformations. These transformations preserve distance. These transformations are rotations, reflections and translations. The rotations form a special group called the special orthogonal group, denoted as, SO(n). The larger orthogonal group, O(n), includes rotations and reflections. An orthogonal map is a linear map T that preserves distances so that d(x,y) = d(Tx,Ty) for all x and y. If the determinant of an orthogonal map is equal to 1 then the linear map T is a rotation. If it is equal to -1 then the linear map T turns space “inside out” as in a reflection.

The group GLn(R) of all invertible linear transformations of Rn and the translation transformation forms a larger group of transformations of the form x # Tx + b, b is a fixed vector and T is an inveetible linear map. The resulting geometry is called affine geometry. Distance and angle are not included in affine geometry. Points, lines and planes after an invertible linear map remain as points, lines and planes and so are part of geometry. There are parallelograms and ellipses are in affine geometry (but not rectangles or circles).

Let G be a group of transformations of Rn, if we are doing G-geometry, then two shapes are equivalent if we can use transformations in G to obtain each other. In G-geometry basic objects form equivalence classes rather than shapes themselves. One can use a continuous deformation to move within an equivalence class. However, the surface of a sphere cannot be deformed into a torus. One does not have a hole and the other does. Invariants, algebraic topology and differential topology deal with these issues.

Spherical geometry occurs on an n-dimensional sphere, Sn, which is the surface of an n+1 dimensional ball of radius 1. When n = 2 then the appropriate group of transformations is SO(3) the group of all rotations about axes that go through the origin. These are symmetries of the sphere S2. On the sphere lines are great circles. Distances are defined along segments of great circles. There are angles defined between these great circle segments. The angles in a spherical triangle add to more than 180 degrees. Two distinct spherical lines in spherical geometry intersect at two points.

The group of transformations that defines hyperbolic geometry is called PSL2(R), the projective special linear group in two dimensions. The special linear group SL2(R) is the set of all 2 by 2 matrices, with row 1 elements of a and b and row two elements of c and d, with determinant ad – bc = 1. To form the projective group one makes this matrix equivalent to the 2 by 2 matrix with row 1 elements of -a and -b and row two elements of -c and -d.

The half plane model of hyperbolic geometry uses the group PSL2(R) andthe upper half-plane of complex numbers. The transformation takes the point z = x + iy and the 2 by 2 matrix above to the point (az + b)/(cz + d). There is only one distance definition that is allowed by this geometry. The Euclid parrallel postulate fails to hold. That is given a line and a point not on the line there more than one line through the point that does not intersect the line. The other Euclidean geometry axioms hold. This means that the parallel axiom cannot be inferred from the other Euclidean axioms. The angles in a hyperbolic triangle always sum to less than 180 degrees. This reflects the fact that the hyperbolic plane has negative curvature, whereas the Euclidean plan is flat (zero curvature).

There are also disk models and hyperboloid models of hyperbolic geometry. The disk model is defined on the open unit disk. The hyperboloid model is the revolution of a hyperbola about the axis. The geometry transformation involve hyperbolic rotations defined by the 2 by 2 matrix with row 1 elements of cosh(q) and sinh(q) and row 2 elements of sinh(q) and cosh(q). The three hyperbolic geometry models are all equivalent.